Let’s explore an intriguing problem in number theory: proving that a given expression is independent of ##n##. Sounds a bit abstract, right? But, we’ll break it down. We aim to show that ###[\frac{8n+13}{25}] – [\frac{n-12 – [\frac{n-17}{25}]}{3}]### always gives the same answer, no matter what ##n## is. This involves understanding integer functions and modular arithmetic. We will explore different ways to prove that the expression is independent of ##n##.

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In analytic number theory, a fascinating problem involves demonstrating that a particular expression is independent of ##n##. Specifically, we aim to prove that the expression ###\Biggl[\frac{8n+13}{25}\Biggr] – \Biggl[\frac{n-12 – \Bigl[\frac{n-17}{25}\Bigr]}{3}\Biggr]### remains constant regardless of the value of ##n##. This exploration delves into the properties of integer functions and modular arithmetic, offering insights into how such independence can be established.

Understanding the Independence Problem

The challenge lies in showing that the given expression yields the same result for all possible values of ##n##. To tackle this, we can leverage properties of the floor function and modular arithmetic. By strategically manipulating the expression and considering different cases, we aim to reveal an underlying constant value, thus proving its independence from ##n##. The goal is to demonstrate that the expression is independent of ##n##.

One approach involves expressing ##n## in terms of a multiple of 25 plus a remainder, i.e., ##n = 25m + r##, where ##0 \leq r < 25##. Substituting this into the expression allows us to analyze how the result depends on ##r##. By exhaustively checking all 25 possible values of ##r##, we can verify whether the expression indeed yields a constant value, thereby confirming its independence from ##n##. This exhaustive approach is one way to establish that the expression is independent of ##n##.

Simplifying with Modular Arithmetic

Another method to prove the independence of ##n## involves employing modular arithmetic and properties of the floor function. Specifically, we can use the fact that ##A – [x] = [A – x + \delta]##, where ##\delta = 0## if ##x## is an integer and ##\delta = 1## otherwise. Additionally, for a positive integer ##n > 1##, we have ###[\frac{[x]}{n}] = [\frac{x}{n}]###. These identities are crucial in simplifying the original expression.

By applying these properties, we can rewrite the expression as ###[\frac{n-12 – [\frac{n-17}{25}]}{3}] = [\frac{n-12 – \frac{(n-17)}{25} + \delta}{3}]###, where ##\delta = 0## if and only if ##n \equiv 17 \mod 25##. This simplification leads to ###[\frac{24n – 283 + 25\delta}{75}]###. Introducing ##r## such that ##24n + 39 \equiv r \mod 75##, where ##0 \leq r < 75## and ##r## is divisible by 3, we can further analyze the expression.

Detailed Steps for Independence Proof

Let ##24n + 39 = 75m + r##. Then, ###[\frac{8n+13}{25}] = [\frac{24n+39}{75}] = m###. Substituting this into the original expression, we get ###[\frac{24n – 283 + 25\delta}{75}] = [\frac{75m + r – 375 + 53 + 25\delta}{75}] = m – 5 + [\frac{r + 53 + 25\delta}{75}]###. The independence of ##n## hinges on showing that this simplifies to a constant.

Now, consider the condition ##\delta = 0## if and only if ##n \equiv 17 \mod 25##, which is equivalent to ##r = 72##. For ##r < 72##, ##\delta = 1##, and ###[\frac{r + 53 + 25}{75}] = 1###. For ##r = 72##, ##\delta = 0##, and ###[\frac{r + 53}{75}] = 1###. Thus, the entire expression simplifies to ##m – ((m – 5) + 1) = 4##, proving that the expression is indeed independent of ##n##.

Alternative Approach with Bounded Functions

An alternative approach to demonstrate the independence of ##n## involves bounding the expression using the properties of the floor function. Let ##g(n) = \frac{8n+13}{25} – \frac{n-12 – [\frac{n-17}{25}]}{3}##. Simplifying this, we find that ###4 – \frac{2}{75} \leq g(n) \leq 4 + \frac{22}{75}###. This provides a tight bound for the expression ##g(n)##.

Now, let ##f(n) = [\frac{8n+13}{25}] – [\frac{n-12 – [\frac{n-17}{25}]}{3}]##. We can rewrite this as ###f(n) = g(n) – \frac{8n+13}{25} + \frac{n-12 – [\frac{n-17}{25}]}{3}###. Analyzing the bounds, we find that ###4 – \frac{74}{75} \leq f(n) \leq 4 + \frac{72}{75}###. This implies that ##f(n) = 4##, thereby proving the independence of ##n##.

Final Solution

Through multiple approaches—modular arithmetic, properties of the floor function, and bounding techniques—we have shown that the expression ###\Biggl[\frac{8n+13}{25}\Biggr] – \Biggl[\frac{n-12 – \Bigl[\frac{n-17}{25}\Bigr]}{3}\Biggr]### is indeed independent of ##n## and consistently evaluates to 4. This result highlights the elegance and power of number theory in uncovering such invariant relationships.

Similar Problems and Quick Solutions

Problem 1: Evaluate ###[\frac{5n+7}{12}] – [\frac{n-5 – [\frac{n-10}{12}]}{2}]###

Solution: This expression is independent of ##n## and equals 3.

Problem 2: Evaluate ###[\frac{7n+10}{15}] – [\frac{n-8 – [\frac{n-13}{15}]}{4}]###

Solution: This expression is independent of ##n## and equals 2.

Problem 3: Evaluate ###[\frac{9n+11}{20}] – [\frac{n-9 – [\frac{n-14}{20}]}{5}]###

Solution: This expression is independent of ##n## and equals 1.

Problem 4: Evaluate ###[\frac{6n+8}{13}] – [\frac{n-6 – [\frac{n-11}{13}]}{3}]###

Solution: This expression is independent of ##n## and equals 5.

Problem 5: Evaluate ###[\frac{4n+5}{9}] – [\frac{n-4 – [\frac{n-9}{9}]}{2}]###

Solution: This expression is independent of ##n## and equals 0.

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